In the framework of quantum mechanics, the wave function is a mathematical description that represents the state of a quantum system. It provides information about the probabilities of different outcomes when measurements are made on the system. The wave function is typically denoted by the symbol Ψ (psi) and is a function of the coordinates of the particles in the system.
In the context of space-time emergence from quantum mechanics, the precise nature of the wave function and its dependencies can vary depending on the specific theoretical approach. However, in most interpretations, the wave function is still a function of the coordinates of the particles in the system. It describes the probabilities of different configurations of the particles, even if the concept of space-time itself is emergent.
The idea of quantum mechanics without space-time refers to theoretical frameworks that attempt to describe quantum phenomena without assuming the existence of a pre-existing, fixed space and time. Such frameworks often involve the idea that space-time itself, including its geometry, arises as an approximate description or an effective concept emerging from more fundamental quantum mechanical principles.
One example of a theory attempting to describe quantum mechanics without space-time is called "quantum graphity." In this approach, the wave function represents the state of a network or graph, and the evolution of the wave function describes the rearrangement and connections of nodes in the graph. The concept of continuous space-time is not fundamental in this framework but emerges as a coarse-grained description at larger scales.
It's worth noting that the concept of space-time emergence from quantum mechanics is an active area of research, and there are various approaches and theoretical frameworks being explored. The precise nature of the wave function and its dependencies in these theories is still a subject of investigation and debate among physicists and researchers in the field.